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EQUATIONS, EQUATIONS, INEQUALITIES & INEQUALITIES &
ABSOLUTE VALUEABSOLUTE VALUE
2
CONTENT
2.1 Linear Equation
2.2 Quadratic Expression and Equations
2.3 Inequalities
2.4 Absolute value
2.1:2.1:
Linear Linear EquationsEquations
4
Objectives
• At the end of this topic, you should be able to• Define linear equations • Solve a linear equation• Solve equations that lead to linear equations • Solve applied problems involving linear
equations
5
Equation in one variable
A statements in which 2 expressions (sides) at least one containing the variable are equal
It may be TRUE or FALSE depending on the value of the variable.
The admissible values of the variable (those in the domain of the variable), if any, that result in a TRUE statement are called solutions or root.
To solve an equation means to find all the solutions of the equation
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Equation in one variable, cont…
An equation will have only one solution or more than one solution or no real solutions or no solution
Solution set – the set of solutions of an equation, {a}
Identity – An equation that is satisfied for every value of the variable for which both sides are defined
Equivalent equations – Two or more equations that have the same solution set.
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Linear Equations
A Linear Equation in one variable is equivalent to an equation of the form
where a and b are real numbers and
The linear equation has the single solution given by the formula
Simplify the given equations first, to solve a linear equations
0ax b 0a
bx
a
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Steps for Solving a Linear Equation
STEP 1: If necessary, clear the equation of fractions by multiplying both sides by the least common multiple (LCM) of the denominators of all the fractions.
STEP 2: Remove all parentheses and simplify
STEP 3: Collect all terms containing the variable on one side and all remaining terms on the other side.
STEP 4: Check your solution (s)
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Solve a Linear Equation
Solve the following equations
1. 3 4 2. 2 6 3
1 1 3 1 13. 5 4 2 1 4. 2
2 3 2 2 2
x x t t
x x y y
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Solve equations that lead to linear equations
Solve the following equations
5 3 3 1 71. 3.
2 1 2 1 1 2
4 52. 5 4. 2 1 1 5 2 5
y 2
x x x x x x
y y y yy
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An equation with no solution
Solve the following equations
3 31. 2
1 1
3 1 2 32.
2 2
x
x x
x x
x x
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Translating Written/Verbal Information into a Mathematical Model
Addition Subtraction Multiplication Division Equals
And From Of Into Is
Plus Subtract Times Over Equals
More Less Product Divided by Same as
Added to Fewer By Quotient of Makes
Together with Minus Percent of Ratio of Leaves
Sum Difference Multiplied by a is to b Yields
Total Take away per Equivalent
Increased by Decreased by
Results in
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Solve applied problems involving linear equations
Example 1 A total of Rp.45.000.000 is invested, some
in stocks and some in bonds. If the amount invested in bonds is half that invested in stocks, how much is invested in each category?
2.2:2.2:
Quadratic Quadratic Expression Expression &Equations&Equations
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Objectives
• At the end of this topic you should be able to• Define quadratic expressions and equations
• Solve quadratic equations by factorization, square root method, and quadratic formula
• Recognize the types of roots of a quadratic equation based on the value of discriminant
• Solve applied problems involving quadratic equations
16
Quadratic Equations
A Quadratic Equation in is an equation equivalent to one of the form
where a, b and c are real numbers and
A Quadratic Equation in the form
is said to be in standard form
3 ways to solve quadratic equations a. Factoring
b. Square root method
c. Quadratic Formula
2 0ax bx c 0a
2 0ax bx c
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Solve a Quadratic Equation by Factoring Solve the following equations
Repeated Solution / root of multiplicity 2 / double root When the left side, factors into 2 linear
equations with same solution
2 2
2 2
1. 5 6 0 3. 9 6 1 0
2. 2 3 4. 3 5 2 0
x x x x
x x x x
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Solve a Quadratic Equation by the Square Root Method
Solve the following equations
221. 5 2. 2 16 x x
2If and 0, then x p p x p
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Solve a Quadratic Equation by the Quadratic Formula Use the method of completing the square to obtain a
general formula for solving the quadratic equation
Solve the following equations
22 4
From 0 to 2
b b acax bx c x
a
2 2
2 2
1. 6 16 0 3. 2 8 5 0
2. 5 4 0 4. 2 3 24 0
x x x x
x x x x
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Discriminant of a Quadratic Equation
For a Quadratic Equation
If there are two unequal real solutions
If there is a repeated solution, a root of
multiplicity 2
If there is no real solution (complex roots)
2 4 0b ac
2 0ax bx c
2 4 0b ac
2 4 0b ac
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Examples
Find a real solutions, if any, of the following equations
2 2
22
2
1. 3 5 1 0 3. 3 4 2 0
25 3 22. 30 18 0 4. 9 0
2
5. 1 0 6. 1 7
x x x x
x xx x
x x x x
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Application of Quadratic Equations
Example 1 The quadratic function
models the percentage of the U.S. population f (x), that was foreign-born x years after 1930. According to this model, in which year will 15% of the U.S. population be foreign-born?
20.0049 0.361 11.79f x x x
2.3:2.3:
Inequalities Inequalities
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Objectives
• At the end of this topic you should be able to• Relate the properties of inequalities• Define and Solve linear inequalities• Define Solve quadratic inequalities• Understand and solve rational inequalities
involving linear and quadratic expression
Properties of Inequalities
1. If a < b and b < c then a < c2. If a < b and c is any number, then a + c < b + c3. If a < b and c is any number, then a – c < b – c 4. If a > 0 and b > 0 then a + b > 05. If a > 0 and b > 0 then ab > 06. If a < b then b – a > 0 7. If a > b and –a < –b8. If a < b and –a > –b9. If a < b and c > 0 then ac < bc 10. If a < b and c < 0 then ac > bc
11.
12. reciprocal property
13. reciprocal property
2 0
1If 0 then 0
1If 0 then 0
a
aa
aa
Solving Linear Inequalities
Solve the following inequality and graph the solution set
1
1. 3 2 5 4. 5 3 2 1
3 52. 4 7 2 3 5. 1 9
2
1 13. 9 5 2 1 6. 4 1 0
3 4
x x
xx x
x x x x
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Solves problems involving linear inequalities
At least, minimum of, no less than
At most, maximum of, no more than
Is greater than, more than
Is less than, smaller than
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Examples
Sasha’s grade in her math course is calculated by the average of four tests. To receive an A for this course, she needs an average at least 89.5. If her current test scores are 84, 92, and 94, what range of scores can she make on the last test to receive an A for the course?
A painter charges RM80 plus RM1.50 per square foot. If a family is willing to spend no more than RM500, then what is the range of square footage they can afford?
Solving Quadratic inequalities
Step 1 - solve the related quadratic equation
Step 2 – plot the solution on a number line
Step 3 – Choose a test number from each interval & substitute the number into the inequality If the test number makes the inequality true
All numbers in that interval will solve the inequality If the test number makes the inequality false
No numbers in that interval will solve the inequality
Step 4 – State the solution set of the inequality ( It is a union of all intervals that solves the inequality) If the inequality symbols are or , then the values
from Step 2 are included. If the symbols are > or <, they are not solutions
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Examples
Solve the following inequality and graph the solution set
22
22
1. 6 0 3. 1 2
2. 3 0 4. 1 2
x x x
x x x
Solving rational inequality
STEP 1: Solve the related equation
STEP 2: Find all values that make any denominator equal to 0
STEP 3: Plot the number found in Step 1 and 2 on a number line
STEP 4: Choose a test number from each interval and determine whether it solves the inequality.
STEP 5: The solution set is the union of all regions whose test number solves the inequality. If the inequality symbol is or , includes the values found in step 1
STEP 6: The solution set never includes the values found in Step 2 because they make the denominator equal to 0
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Examples
Solve the following inequality and graph the solution set
4 1 21. 0 3.
2 1 1
5 22. 4 4. 3
1 4
x
x x x
x x
x x
2.4:2.4:
Absolute Value Absolute Value
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Objectives
• At the end of this topic you should be able to• Define absolute value• Understand, state and use the properties of
absolute value• Solve problems on equations and inequalities
involving absolute value
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What is Absolute Value
The absolute value can be define as:
, 0
, 0
a aa
a a
The absolute value represents the distance of a point on the number line from the origin
a
- a
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Properties of Absolute Value
1. 0
2.
3.
4.
5.
6. , 0
a
a a
a b b a
a b b a
ab a b
aab
b b
For any real number a and b
Properties of Absolute Value
Equations involving absolute value
If is a positive real number and if is any algebraic expression, then
is equivalent to or
a y
y a y a y a
Inequalities involving absolute value
If a is positive real number and if is any algebraic expression, then
is equivalent to
is equivalent to
In other words, is equivalent to and
is equivalent to
a y
y a a y a
y a a y a
y a a y y a
y a y
and
is equivalent to and
a y a
u a u a u a
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Solve equations involving absolute value
Solve the following equation
2
11. 4 13 3. 5 =1
2
2. 3 9 4. 16 =0
x x
x x
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Solve inequalities involving absolute value
Solve the following inequalities. Graph the solution set
1. 2 4 3 3. 3
2. 1 4 <5 4. 2 5 3
x x
x x
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Application of Absolute value
The inequality
describes the percentage of children in the population who think that being grounded is a bad thing about being kid. Solve the inequality and interpret the solution
9 2.9x
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